Zhu's connection of these parameters to bounded expansion in 2009 [1]} renewed interest in analyzing and utilizing them.
Reidl et al. used the weak coloring numbers to prove that neighborhood complexity characterizes bounded expansion [2]}, while Nadara et al. initiated an empirical study of algorithmic techniques used to compute the generalized coloring numbers in practice [3]}.
The characterization also led to new algorithms for bounded expansion;
Dvořák gave a constant factor approximation for Dominating Set  [4]}, and Reidl and Sullivan gave an exact FPT algorithm for Subgraph Counting  [5]}.
Both of these algorithms rely on computing a linear order of the vertices which witnesses a small weak \(r\) -coloring number for some fixed radius \(r\) . Since Grohe et al. showed that determining the weak \(r\) -coloring number is NP-hard when \(r \ge 3\)  [6]}, both algorithms use an approximation. Dvořák gave a parameterized constant-factor approximation for the weak \(r\) -coloring number in \(O(rn^3)\)  time through an approximation for \(r\) -admissibility [4]}.
